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A156984 Triangle T(n, k) = n!*Sum_{j=k..n} (-1)^(j+k)*binomial(k+j, j)/j!, read by rows. 2
1, 0, 2, 1, 1, 6, 2, 7, 8, 20, 9, 23, 47, 45, 70, 44, 121, 214, 281, 224, 252, 265, 719, 1312, 1602, 1554, 1050, 924, 1854, 5041, 9148, 11334, 10548, 8142, 4752, 3432, 14833, 40319, 73229, 90507, 84879, 63849, 41019, 21021, 12870, 133496, 362881, 659006, 814783, 763196, 576643, 364166, 200629, 91520, 48620 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row sums are: {1, 2, 8, 37, 194, 1136, 7426, 54251, 442526, 4014940, ...}.

The first column gives the sub-factorials, or rencontres, numbers A000166. See Riordan's p_n(k) equation 17 for further reference.

REFERENCES

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 57-65

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k) = n!*Sum_{j=k..n} (-1)^(j+k)*binomial(k+j, j)/j!.

EXAMPLE

Triangle begins as:

       1;

       0,      2;

       1,      1,      6;

       2,      7,      8,     20;

       9,     23,     47,     45,     70;

      44,    121,    214,    281,    224,    252;

     265,    719,   1312,   1602,   1554,   1050,    924;

    1854,   5041,   9148,  11334,  10548,   8142,   4752,   3432;

   14833,  40319,  73229,  90507,  84879,  63849,  41019,  21021, 12870;

  133496, 362881, 659006, 814783, 763196, 576643, 364166, 200629, 91520, 48620;

MAPLE

A156984:= (n, k) -> add( (-1)^(j+k)*binomial(k+j, j)*(n!/j!), j=k..n );

seq(seq(A156984(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 09 2021

MATHEMATICA

Table[n!*Sum[(-1)^(j-k)*Binomial[k+j, j]/j!, {j, k, n}], {n, 0, 12}, {k, 0, n}]//Flatten

PROG

(Sage) flatten([[sum( (-1)^(j+k)*binomial(n, j)*binomial(k+j, j)*factorial(n-j) for j in (k..n) ) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021

(Magma) [(&+[ (-1)^(j+k)*Binomial(n, j)*Binomial(k+j, j)*Factorial(n-j): j in [k..n]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

CROSSREFS

Cf. A000166.

Sequence in context: A179972 A085826 A112477 * A181621 A307070 A321615

Adjacent sequences:  A156981 A156982 A156983 * A156985 A156986 A156987

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 20 2009

EXTENSIONS

Edited by G. C. Greubel, Mar 09 2021

STATUS

approved

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Last modified June 26 12:19 EDT 2022. Contains 354882 sequences. (Running on oeis4.)